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arXiv · 2412.07164

Order Polytopes of Dimension $\leq 13$ are Ehrhart Positive

Feihu Liu·Guoce Xin·Zihao Zhang
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Abstract

The order polytopes arising from the finite poset were first introduced and studied by Stanley. For any positive integer $d\geq 14$, Liu and Tsuchiya proved that there exists a non-Ehrhart positive order polytope of dimension $d$. They also proved that any order polytope of dimension $d\leq 11$ is Ehrhart positive. We confirm that any order polytope of dimension $12$ or $13$ is Ehrhart positive. This solves an open problem proposed by Liu and Tsuchiya. Besides, we also verify that any $h^{*}$-polynomial of order polytope of dimension $d\leq 13$ is real-rooted.

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Feihu Liu, Guoce Xin, Zihao Zhang. 2024-12-10. Order Polytopes of Dimension $\leq 13$ are Ehrhart Positive. https://doi.org/10.1016/j.disc.2025.114833

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